Conference napalm::guitar

2304.0. "Spkr Cab Dimensions---How To Roll Your Own" by RGB::ROST (Fart Fig Newton) Mon Aug 26 1991 14:18

    Here's a note from USENET that has what you have all been waiting for,
    the equations for calculating speaker cabinet dimensions based on the
    speaker specs.  These are the dreaded "Thiele-Small" equations that
    everybody always talks about.  
    
    							Brian

    Article        16949
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From: bill@thd.tv.tek.com (William K. McFadden)
Newsgroups: comp.sys.handhelds,comp.sys.hp48,rec.audio,rec.audio.car,rec.music.makers
Subject: Speaker Design Equations
Message-ID: <1991Aug23.180107.3863@tvnews.tv.tek.com>
Date: 23 Aug 91 18:01:07 GMT
Sender: news@tvnews.tv.tek.com (news user)
Organization: Tektronix TV Products
Lines: 143
Originator: bill@thd.tv.tek.com
 
I recently posted a note that the new version of my speaker design
library for the HP48 had been posted to comp.sources.hp48.  I have
since discovered that many people who don't have the HP48 would still
like to have the equations.
 
Most of the equations came from the following technical paper:
 
G. Margolis and R. H. Small, "Personal Calculator Programs for
Approximate Vented-Box and Closed-Box Loudspeaker System Design," J.
Audio Eng. Soc., vol. 29, pp. 421-441 (1981 June); pp. 824 (1981
Nov.).
 
Others came or were derived from the following papers:
 
A.N. Thiele, "Loudspeakers in Vented Boxes, Parts I and II," J.  Audio
Eng. Soc., vol. 19, pp. 382-392 (1971 May); pp. 471-483 (1971 June).
 
R.H. Small, "Direct-Radiator Loudspeaker System Analysis," J.  Audio
Eng. Soc., vol. 20, pp. 383-395 (1972 June).
 
R.H. Small, "Closed-Box Loudspeaker Systems," J. Audio Eng.  Soc., vol.
20, pp. 798-808 (1972 Dec.); vol. 21, pp. 11-18 (1973 Jan./Feb.).
 
R.H. Small, "Vented-Box Loudspeaker Systems," J. Audio Eng.  Soc., vol.
21, pp. 363-372 (1973 June); pp. 438-444 (1973 July/Aug.); pp.  549-554
(1973 Sept.); pp. 635-639 (1973 Oct.).
 
 
Here are my equations.  All units are in the SI (mks) system.  ^
denotes exponentiation.  LOG() is base 10.
 
 
VARIABLE DEFINITIONS:
 
Vas	Volume of air having same acoustic complaince as driver suspension
Qts	Total driver Q at Fs
Fs 	Resonant frequency of driver
PEmax	Thermally-limited maximum RMS input power
SPL	Efficiency of driver in dB SPL at 1W/1m
Dia	Diameter of driver
xmax	Peak displacement limit of driver diaphragm (1/2 of "throw")
Vb 	Inside volume of enclosure
Fb 	Resonance frequency of enclosure
F3dB	Half-power (-3 dB) frequency of loudspeaker system response
Fmax	Upper frequency limit of driver's piston range
dBpeak	Maximum peak or dip of loudspeaker system response
Par	Estimated displacement-limited acoustic power rating
Per	Estimated displacement-limited electrical power rating
\Gno	Percent driver efficiency (\Gn is greek character eta)
PeakSPL	Thermally-limited RMS sound pressure level in passband
Sd 	Estimated effective projected surface area of driver diaphragm
Vd 	Peak displacement volume of driver diaphragm
K1 	Power rating constant
K2 	SPL rating constant
 
The following additional variables are defined for the closed box case:
 
Qb	Total Q of system at Fb
Amax	Maximum amplitude of loudspeaker frequency response
Vr	Ratio of Vas to Vb
Qr	Ratio of Qb to Qts and Fb to Fs
 
The following additional variables are defined for the ported box case:
 
Dmin	Minimum diameter of tubular vent to prevent excessive vent noise
Dv	Diameter of tubular vent
Lv	Length of tubular vent
 
For the ported box case, the following apply:
 
1. Fb is the tuning frequency for the vent.
2. To use a square vent, enter the vent width times 1.13 or
   [2/SQRT(pi)] for Dv.
 
 
CLOSED BOX DESIGN:
 
Vb = Vas/Vr
Fb = Qr/Qts
F3dB = Qr*Fs*((1/Qb^2-2+((1/Qb^2-2)^2+4)^0.5)/2)^0.5
Fmax = c/(pi*0.83*Dia)
dBpeak = 20*LOG(Amax)
Par = K1/Amax^2
Per = Par/(\Gno)
\Gno = 10^((SPL-112)/10)
PeakSPL = SPL+10*LOG(PEmax)
Sd = pi*(Dia*0.83)^2
Vd = Sd*xmax
Amax = Qb^2/(Qb^2-0.25)^0.5 for Qb >1/2^0.5, 1 otherwise
K1 = (4*pi^3*Ro/c)*Fb^4*Vd^2
K2 = 112+10*LOG(K1)
Vr = Qr^2-1
Qr = (1/Qts)/(1/Qb-0.1)
 
Frequency-dependant equations:
Fr = (F/Fb)^2
dBmag = 10*LOG(Fr^2/((Fr-1)^2+Fr/Qb^2))
Pmax = K1*((Fr-1)^2+Fr/Qb^2))/(\Gno)
SPLmax = K2+40*LOG(F/Fb)
Thermally-limited RMS SPL = PeakSPL+dBmag
 
 
PORTED BOX DESIGN:
 
Vb = 20*Qts^3.3*Vas
Fb = (Vas/Vb)^0.31*Fs
F3dB = (Vas/Vb)^0.44*Fs
Fmax = c/(pi*0.83*Dia)
dBpeak = 20*LOG(Qts*(Vas/Vb)^0.3/0.4)
Par = 3*F3dB^4*Vd^2
Per = Par/(\Gno)
\Gno = 10^((SPL-112)/10)
PeakSPL = SPL+10*LOG(PEmax)
Dmin = (Fb*Vd)^0.5
Lv = 2362*Dv^2/(Fb^2*Vb)-0.73*Dv
Sd = pi*(Dia*0.83)^2
Vd = Sd*xmax
K1 = (4*pi^3*Ro/c)*Fs^4*Vd^2
K2 = 112+10*LOG(K1)
 
Frequency-dependent equations:
Fn2 = (F/Fs)^2
Fn4 = Fn2^2
A = (Fb/Fs)^2
B = A/Qts+Fb/(7*Fs)
C = 1+A+(Vas/Vb)+Fb/(7*Fs*Qts)
D = 1/Qts+Fb/(7*Fs)
E = (97/49)*A
dBmag = 10*LOG(Fn4^2/((Fn4-C*Fn2+A)^2+Fn2*(D*Fn2-B)^2))
Pmax = (K1/\Gno)*((Fn4-C*Fn2+A)^2+Fn2*(D*Fn2-B)^2)/(Fn4-E*Fn2+A^2)
SPLmax = K2+10*LOG(Fn4^2/(Fn4-E*Fn2+A^2))
Thermally-limited RMS SPL = PeakSPL+dBmag
 
 
CONSTANTS:
 
c = speed of sound in air (345 m/s)
pi = 3.14159265359
Ro = density of air (1.18 kg/m^3)
-- 
Bill McFadden    Tektronix, Inc.  P.O. Box 500  MS 58-639  Beaverton, OR  97077
bill@tv.tv.tek.com, ...!tektronix!soul!bill               Phone: (503) 627-6920
"How can I prove I am not crazy to people who are?"